so that is constant with respect to x- that is, where f is an arbitrary function of y only.
Now treat as an ordinary differential equation in y. It is a first order, linear, differential equation so it can be integrated using an "integrating factor"- that is, a function such that .
since we must have or , and .
That is, so that where F(y) is an anti-derivative of f. Since f was arbitrary, F is any arbitrary (differentiable) function of y. However, since we have been doing this with y only, treating x as a constant, that "constant of integration", C, may be an arbitrary function of x-
so where F and G can be any arbitrary differentiable functions.
(Yes, just as the solutions to ordinary differential equations involve unknown constants, so the solutions to partial differential equations involve unknown functions).
Now, use the additional conditions (of which you don't have enough):
so . Now you have but you still need another condition, perhaps something of the form u(0,y)= f(y), to determine F(y).